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Average propensity to consume (APC) is the part of income that is consumed.

$\mathrm{APC}=\frac{\mathrm{Consumption}}{\mathrm{Income}}$

For example, if the national income is $100 billion, the consumption is $90 billion. The APC is 0.9 ($90 billion/$100 billion = 0.9) or 90%.

Marginal propensity to consume (MPC) is the rate at which the change in total income is consumed.

$\mathrm{MPC}=\frac{â–³\mathrm{Consumption}}{â–³\mathrm{Income}}$

For example, suppose the income increases by $1000 and consumption increases by $900, the MPC is 0.9 ($900/$1000 = 0.9)

Marginal propensity to consume is the fraction of change in consumption out of change in income. It depicts the rate by which a change in income is consumed.

$\mathrm{MPC}=\frac{â–³\mathrm{Consumption}}{â–³\mathrm{Income}}$

On the contrary, marginal propensity to save is the ratio of change in savings to income change. It demonstrates the rate by which a change in income is saved.

MPS = â–³ Savings

â–³ Income

Since the income is split into consumption and saving, the amount once consumed cannot be saved anymore. Thus, MPC and MPS are mutually exclusive; the same part of income cannot be saved and consumed.

The fraction of any change in income that is not consumed is naturally saved. The fraction of change in income consumed and the fraction of change in income saved must exhaust the total change in income. Therefore, the sum of MPC and MPS should be equal to one.

MPS + MPC = 1

Mathematical equations can help to express the fundamental reason. Since income is equal to the summation of consumption and saving, change in income also equals the sum of the change in consumption and change in saving.

$$\begin{gathered} Y=C+S \\ â–³Y=â–³C+â–³S \\ \mathrm{On}\mathrm{dividing}\mathrm{the}\mathrm{equation}\mathrm{by}\mathrm{change}\mathrm{in}\mathrm{income} \\ \frac{â–³\mathrm{Y}}{â–³\mathrm{Y}}=\frac{â–³\mathrm{C}}{â–³\mathrm{Y}}+\frac{â–³\mathrm{S}}{â–³\mathrm{Y}} \\ 1=\mathrm{MPC}+\mathrm{MPS} \end{gathered}$$

Therefore, the sum of MPC and MPS is equal to one.